How to go about having a children's book published?

[NumptyNu]

Does anyone know where I start if I have an idea for a story? How do I find an illustrator / publisher? What do I need to think about?

The first thing you need to do is write it. Once you have done that, you find an agent willing to offer it up to publishers. They will find you an illustrator.

That's one way.

Another is to write it, then go straight to the publishers yourself but this way is harder as many won't even look at manuscripts which haven't been sent via a reputable agent.

Agents who represent children's authors can be found in the Writers and Artist's Yearbook which is available in the library...for reference only. Or you can buy one here

You need that as it lists all the agents and publishers dealing with children's literature. Read the entries carefully as some only deal with picture books...others with text books and so on.

Perfect, thanks Neo!

You can self-publish with lulu.com
Or write a book for the kindle?

I self published a young adult ebook with Smashwords
It's easy to upload books, Smashwords has a free "style guide" that says how to format books, and they distribute to B&N, Kobo, apple etc.
I also published with KDP (Kindle Direct Publishing), they have slightly different requirements but are still easy to upload.
Re. Illustrators, mine is a student friend who illustrates perfectly for my style of writing. It's like she can draw what's in my head. Look for pictures you like, and see who draws that way, and if you find your perfect illustrator, keep hold of them!